## Response theory of optical forces in two-port photonics systems: a simplified framework for examining conservative and non-conservative forces |

Optics Express, Vol. 19, Issue 22, pp. 22322-22336 (2011)

http://dx.doi.org/10.1364/OE.19.022322

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### Abstract

We extend the response theory of optical forces to general electromagnetic systems which can be treated as multi-port systems with multiple mechanical degrees of freedom. We demonstrate a fundamental link between the scattering properties of an optical system to its ability to produce conservative or non-conservative optical forces. Through the exploration of two nontrivial two-port systems, including an analytical Fabry-Perot interferometer and a more complex particle-in-a-waveguide structure, we show perfect agreement between the response theory and numerical first-principle calculations. We show that new insights into the origins of optical forces from the response theory provide clear means of understanding conservative and non-conservative forces in a regime where traditional gradient force picture fails.

© 2011 OSA

## 1. Introduction

## 2. Response theory of optical forces in single-port and multi-port optical systems

24. P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express **17**(20), 18116–18135 (2009). [CrossRef] [PubMed]

*external*amplitude and phase responses of an optical system. Specifically, to calculate optical force within any lossless system which satisfies energy and particle conservation, the knowledge of optical responses as a function of mechanical degrees of freedom is sufficient. The mechanical degrees of freedom can include displacement, angle of rotation and other forms. Unlike conventional methods, such as the Maxwell stress tensor [21,22,28] or the Lorentz force density [23

23. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. **2**(1), 021875 (2008). [CrossRef]

29. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**(22), 5375–5401 (2004). [CrossRef] [PubMed]

31. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express **13**(7), 2321–2336 (2005). [CrossRef] [PubMed]

*n*independent degrees of freedom (Fig. 1b), the optical response becomes

_{S˜(q1,q2,⋯qn)}, where

_{qk}is the

*k*-th component of the generalized coordinate

**q**. The corresponding

*k*-th component of the optical force isand the total optical force becomes

**F**is the gradient of a scalar field, any lossless single-port system produces a conservative force field. This observation has been analytically and numerically verified in Ref [24

24. P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express **17**(20), 18116–18135 (2009). [CrossRef] [PubMed]

_{U(q,ω)=−(Pi/ω)ϕ(q,ω)}.

*single*mechanical degree of freedom (Fig. 1c), the optical force takes the form [24

24. P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express **17**(20), 18116–18135 (2009). [CrossRef] [PubMed]

_{ϕjo(q,ω)}is the phase angle of the wave exiting the

*j-*th output port

*q*is bounded in a finite range), the force field can again be described by a scalar optical potential [24

**17**(20), 18116–18135 (2009). [CrossRef] [PubMed]

2. D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef] [PubMed]

35. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**(2), 569–582 (1992). [CrossRef] [PubMed]

36. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**(1), 013602 (2008). [CrossRef] [PubMed]

8. P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics **1**(11), 658–665 (2007). [CrossRef]

16. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics **1**(7), 416–422 (2007). [CrossRef]

25. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. **101**(12), 128301 (2008). [CrossRef] [PubMed]

26. B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **80**(1), 010401 (2009). [CrossRef] [PubMed]

_{∇×∇ϕ=0}and

_{∇×(ϕA)=ϕ∇×A+∇ϕ×A}. For systems with only a single port, the output power is constant (

_{P1=Pi}) and independent of the position

**q**(

_{∇qϕj(q,ω)}. Hence, in general, optical forces are not conservative. More details of non-conservative of optical forces will be examined in Section 4.

_{|a˜1| >0}, and

_{|a˜2| =0}. Thus the output waves are determined only by two elements in the scattering matrix:

_{b˜1(ω,q1,q2)=S˜11(ω,q1,q2)·a˜1,b˜2(ω,q1,q2)=S˜21(ω,q1,q2)·a˜1.}

## 3. An analytical example: optical forces in a Fabry-Perot interferometer

3. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science **321**(5893), 1172–1176 (2008). [CrossRef] [PubMed]

37. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express **15**(25), 17172–17205 (2007). [CrossRef] [PubMed]

*l*, corresponding to a round-trip phase shift

_{0}_{[−rjtjt−r].}

*t*and the reflectivity

*r*are both real numbers [34]. For the scope of this paper, we limit our attention to lossless and reciprocal systems (absent of magneto-optical materials), such that

_{t=1−r2}. Note however, the general results of this section are very broadly applicable to a variety of systems if

*r*and

*t*are taken to be frequency dependent, as the partial mirrors considered here can be easily replaced with more complex structures consisting of photonic crystal slabs [38

38. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**(23), 235112 (2002). [CrossRef]

_{b1=−r−re−jδ1−r2e−jδa1b2=t2e−jδ1−r2e−jδa1a=jt1−r2e−jδa1b=−jtre−jδ1−r2e−jδa1}

*a*and

*b*represent the amplitudes of the right- and left- going waves respectively between the two mirrors. The optical forces acting on Mirrors 1 and 2, can be found by computing the change in photon momentum induced by each mirror:

_{F1=1c(|a1|2+|b1|2−|a|2−|b|2)F2=1c(|a|2+|b|2−|b2|2)}

_{l1+l0+l2=L≡constant}), so that the phases of the waves entering or exiting the two-port system are also taken at a fixed location. Solving the scattering matrix at the two reference planes, we have

_{S11=−r−re−jδ1−r2e−jδe−j2ωl1/c,S21=t2e−jδ1−r2e−jδe−jω(l1+l2)/c.}

*F*on Mirror 1, we take

_{1}*l*to be a constant while

_{2}*l*is varied, yielding

_{0}*S*-matrix and combining with Eq. (4), one arrives at

*l*as a constant:

_{1}*l*, is entirely determined by the position of the chosen mirror. The work done by the optical force is path independent, since only one path exists between any two states of the system. In contrast, if both mirrors are allowed to move, two degrees of freedom are present in the system. For simplicity, we take these to be the cavity length,

_{0}*l*, and the position of the aggregate Fabry-Perot system. Since this two-port system is taken to have two degrees of freedom, a seemingly identical Fabry-Perot system produces

_{0}*non-conservative*forces.

*l*while translating the entire system (as a unit) toward and away from the optical source. For example, while moving the cavity towards the source in its transmissive state, no work is done. Conversely, moving the cavity towards the source in its reflective state, work is performed by light. Hence, numerous state-diagrams can be constructed to yield nonzero mechanical work through a closed path in state-space. Hence, this simple system intuitively demonstrates the non-conservative nature of optical forces in optical systems that feature

_{0},*both*multiple port and multiple degrees of freedom.

## 4. A numerical example: a moving cylinder in a single-mode waveguide

2. D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef] [PubMed]

39. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**(4), 156–159 (1970). [CrossRef]

*a*. The incident wave is TE-polarized (E-fields along z direction), and the incident frequency is below the cutoff of the second-order spatial mode. A free silicon cylinder (

_{ε=12}) scatters the incident wave and experiences an associated optical force. The entire system is uniform along the

**z**direction. Despite the inaccuracies of the gradient force model, we first examine this system using this model, as it is the paradigm through which a great many non-conservative systems have been examined [4

4. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics **4**(4), 211–217 (2010). [CrossRef]

40. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. **94**(10), 4853–4860 (1997). [CrossRef] [PubMed]

4. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics **4**(4), 211–217 (2010). [CrossRef]

**y**), with a force amplitude of

_{Fgrad~α∇I}[40

40. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. **94**(10), 4853–4860 (1997). [CrossRef] [PubMed]

_{α}is the polarizability of the object, and

*I*represents the local intensity of the incident beam. Since the fundamental TE mode possesses an intensity profile independent of frequency (Fig. 3e), the gradient force distribution takes on a sinusoidal profile independent of the frequency (Fig. 3d) [33]. At any location, the gradient force points towards the center of the waveguide. On the other hand, the scattering force is generally proportional to the local intensity and takes on the profile shown in Fig. 3e.

_{λ/10}[23

23. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. **2**(1), 021875 (2008). [CrossRef]

_{.}, and a surface integral over a closed surface surrounding the object yields the total optical force [28]:

_{F→= ∮ST↔⋅dS→.}

_{Pt}and the phase

_{ϕt}of the transmitted wave, the power

_{Pr}of the reflected wave, the axial force

*x*and

*y*. The high degree of symmetry within this system allows us to reduce the total force given by Eq. (6) into its essential components as, where

_{kx=[(ω/c)2−(π/ac)2]1/2}and

_{ω}is the angular frequency of the incident wave. Using a finite-element solver (COMSOL Multiphysics), we numerically evaluated local E and B fields, as well as the power and the phase of the reflection and transmission, for a cylinder radius of r = 0.05

*a*at various lateral positions and under an incident frequency of 1.36 (

*c*/2

*a*) and an incident power of 1W/m. Excellent agreement is found between the RTOF method and the Maxwell stress tensor method for both

_{Faxial}and

_{Flateral}, as can be seen in Fig. 4a and 4b. It is important to emphasize that, with RTOF method, most time-domain solvers allow the efficient extraction of optical forces over a broad range of frequencies from power and phase response obtained in a single simulation with pulsed excitation [38

38. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**(23), 235112 (2002). [CrossRef]

*reflected*power as a function of the cylinder location. This result is in stark contrast with the gradient-force paradigm, which leads one to expect a force which is prescribed by the distribution of the

*incident*power. The axial force is also dependent on the axial k vector of the guided modes, and is consistent with momentum conservation. In other words, a reflected photon acquires a momentum change during scattering, and completely transfers it to the scattering object, regardless of the complexity of the local fields. Consequently, the axial force is bounded, with a maximum value of

_{2Pincidentkx/ω}for a given frequency

_{ω}.

*both*from the reflected and transmitted waves, as seen from the red and blue curves of Figs. 4d and 4e. Interestingly, no apparent upper bound exists because the lateral phase gradient is not tied to the propagation constant and can be engineered to large values using high-Q resonances. Surprisingly, as seen by Fig. 4e, the lateral force is also seen to repel the cylinder from the center of the waveguide where the incident fields reach intensity maximum. This sign reversal from the gradient force prediction is entirely attributed to the phase gradient term within Eq. (12), since the transmitted power and reflected power are both positive. In generating such repulsive forces, the contribution from the reflected wave is comparable to that of the transmitted wave.

*a*geometry explored in Fig. 4, two additional cases with larger cylinder radii: 0.075

*a*and 0.125

*a*are presented in Fig. 5 . The force components are plotted for these three particle radii for frequencies spanning the entire single mode range, from the cut-off frequencies of the 1st-order TE mode to the 2nd-order TE mode. The axial and lateral force components are plotted for positions spanning half of the waveguide, since

*F*is symmetric with respect to the center of the waveguide and

_{axial}*F*is anti-symmetric. Note, the center of the cylinder cannot reach the edge of the waveguide due to its finite radius. Thus, the positions where the geometry of the system prevents motion are shown as gray shaded regions.

_{lateral}_{2Pincidentkx/ω}, increases with frequency, and peaks at 1.15 nN/

*a*, independent of the particle size. With increasing frequency, the lateral force, shows a gradual transition from attraction to repulsion with respect to the waveguide center in the case of r = 0.075

*a.*Multiple attraction and repulsion regions can be seen for a larger radius of r = 0.125

*a*. The region where the high-index scatterer reaches stable lateral locations (experiencing zero force in the lateral direction) is shown by the solid white curve in the force plots Fig. 5b, 5f, and 5i. Note, these behaviors contrast starkly with the gradient force model, through which the equilibrium position is predicted to be at the center of the waveguide, as Figs. 5b, 5f, and 5i reveal vanishing lateral forces at locations which are far away from the waveguide center. Interestingly, since the stable particle locations strongly depends on the size and the dielectric constant of the moving object, this phenomenon can be applied to sorting high-index particles in microfluidic waveguides. Further discrepancy from the gradient force model is seen from the fact that, at frequencies where repellent forces exist (blue regions in Fig. 5f and 5i), the center of the waveguide is an unstable equilibrium location. Hence, in the presence of Brownian motion, the particle is unlikely to be found at this position.

_{∇Pt}and

**y**direction, because both the power and the phase responses are independent of the axial position,

*x*, of the cylinder, due to the translational symmetry. The curl of the force field is therefore simplified as:

_{∇×F→=1ω∇Pr×∇ϕr.}

_{F→∼∇I×∇ϕ}[36

36. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**(1), 013602 (2008). [CrossRef] [PubMed]

*I*and

_{ϕ}are taken from the incident fields. Further simplification can be made by taking into account the relation

_{∂ϕr(x,y)/∂x=2kx}. Here, the curl is found to be

*F*= 0) is not at the waveguide center, the curl of the optical forces is also nontrivial and optical heating occurs. In practice, if one seeks to minimize the positional uncertainty, lower frequency is therefore preferred.

_{lateral}*N*propagating modes, the total port count amounts to

*2N*, since transmission and reflection mediated through each eigenmode are two independent ports. Especially for those systems with analytical solutions, the M-port analysis presented in Section 2 could yield additional physical intuition. In contrast, for open systems, each radiation mode needs to be tracked as an independent port, for example, through a near-field to far-field transformation [42]. RTOF theory may provide some simplification for special cases with known analytical solutions.

## 5. Conclusion

## Acknowledgement

## References and links

1. | A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. |

2. | D. G. Grier, “A revolution in optical manipulation,” Nature |

3. | T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science |

4. | D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics |

5. | K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics |

6. | M. L. Povinelli, S. G. Johnson, M. Lonèar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express |

7. | M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. |

8. | P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics |

9. | M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature |

10. | M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics |

11. | M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. |

12. | Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. |

13. | G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature |

14. | M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature |

15. | Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics |

16. | M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics |

17. | A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature |

18. | L. Novotny, “Forces in Optical Near-Fields,” in |

19. | T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. |

20. | T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. |

21. | P. Penfield and H. A. Haus, |

22. | J. D. Jackson, |

23. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. |

24. | P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express |

25. | Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. |

26. | B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express |

28. | L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, |

29. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express |

30. | M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express |

31. | A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express |

32. | H. A. Haus, |

33. | D. M. Pozar, |

34. | H. A. Haus, “Mirrors and Interferometers,” in |

35. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

36. | Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. |

37. | T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express |

38. | S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B |

39. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

40. | A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. |

41. | W. Greiner, |

42. | A. Taflove and S. C. Hagness, |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(140.7010) Lasers and laser optics : Laser trapping

(200.4880) Optics in computing : Optomechanics

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optomechanics

**History**

Original Manuscript: August 23, 2011

Revised Manuscript: October 13, 2011

Manuscript Accepted: October 14, 2011

Published: October 24, 2011

**Virtual Issues**

Vol. 6, Iss. 11 *Virtual Journal for Biomedical Optics*

Collective Phenomena (2011) *Optics Express*

**Citation**

Zheng Wang and Peter Rakich, "Response theory of optical forces in two-port photonics systems: a simplified framework for examining conservative and non-conservative forces," Opt. Express **19**, 22322-22336 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-22322

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### References

- A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron.6(6), 841–856 (2000). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature424(6950), 810–816 (2003). [CrossRef] [PubMed]
- T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321(5893), 1172–1176 (2008). [CrossRef] [PubMed]
- D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics4(4), 211–217 (2010). [CrossRef]
- K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011). [CrossRef]
- M. L. Povinelli, S. G. Johnson, M. Lonèar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express13(20), 8286–8295 (2005). [CrossRef] [PubMed]
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